Problem 105 - Algebra 1

Question

Use a graph to solve the linear system. Check your solution algebraically. (Review 7.1 ) $$\begin{aligned}&-3 x+4 y=-5\\\&4 x+2 y=-8\end{aligned}$$

Step-by-Step Solution

Answer

The point of intersection determined by the graph gives the solution to the system of equations. However, the specific solution cannot be given without visualization of the graph. After obtaining the point from the graph, verify it by substitution into the original equations.

Step 1: Graph the Equations

Using the general form of the equation of a line \( ax+by=c \), consider both equations to be in the form \( y = mx + b \). The first equation, by solving for y, yields \( y = (3/4)x + 5/4 \) and the second equation yields \( y = -2x + 4 \). These equations can be graphed on the xy-plane by plotting their y-intercepts and using their slopes to get more points.

Step 2: Determine the Point of Intersection

The solution of the system of equations will be the point where the two lines intersect. By visually observing the plotted graphs, identify the point of intersection and note down its coordinates (x, y).

Step 3: Check the Solution Algebraically

To ensure the solution is correct, substitute the coordinates of the point of intersection into the original equations respectively to confirm they satisfy them. Both equations should be true when the identified coordinates are substituted for x and y.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in understanding linear systems. When you have equations like \(-3x + 4y = -5\) and \(4x + 2y = -8\), you can convert them into a format that's easy to graph. This form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the first equation, rearranging gives \( y = \frac{3}{4}x + \frac{5}{4} \), indicating a slope of \(\frac{3}{4}\) and a y-intercept at \(\frac{5}{4}\). The second equation simplifies to \(y = -2x + 4\), with a slope of \(-2\) and a y-intercept at \(4\).

• Start by plotting the y-intercept on the graph, as this is where the line crosses the y-axis.
• Use the slope to find another point. For example, a slope of \(\frac{3}{4}\) means you rise 3 units up for every 4 units you move to the right.
• Connect the dots to draw your line.

Graph both equations on the same xy-plane. Carefully plotting the lines will visually show you where, or if, they intersect.

Point of Intersection

The point of intersection is a critical concept in solving linear systems. It represents the solution to the system of equations, where both lines cross at the same point.To find the point of intersection, look where the two graphed lines meet. This point will have coordinates \((x, y)\) that satisfy both equations simultaneously. In our case, identifying this point by observing the graph is straightforward. Make sure you have both lines accurately drawn for precision in determining the exact intersection point. If the graphing is done correctly, you can easily visually find the intersection point.The exactness of the point depends on how precise you were with plotting points and drawing lines. If unsure, using algebraic methods to verify your visual answer is a great follow-up.

Solving Equations Algebraically

Solving equations algebraically provides a way to confirm the graphically determined intersection point of a linear system. Once you've visually identified the intersection point, substitute its coordinates back into the original equations. This verifies whether the point is an actual solution to both equations.For example, suppose you found the point \((x, y)\). Substitute \(x\) and \(y\) into each original equation:

• Check the first equation: \(-3x + 4y = -5\).
• Check the second equation: \(4x + 2y = -8\).

Both sides of these equations should equal when simplified if \((x, y)\) is correct.This method not only confirms the intersection point but also reinforces understanding of solving linear equations by substitution. It is a reliable way to ensure the point found on the graph is indeed the solution.